arXiv:hep-lat/0510043 · Proceedings of XXIIIrd International Symposium on Lattice Field Theory — PoS(LAT2005)

Unitary Evolution on a Discrete Phase Space

Stam Nicolis, Emmanuel G. Floratos

Published Dec 14, 2005• 1 citation

Abstract

We construct unitary evolution operators on a phase space with power of two discretization.These operators realize the metaplectic representation of the modular group SL(2, Z 2 n ).It acts in a natural way on the coordinates of the non-commutative 2-torus, T 2 2 n and thus is relevant for noncommutative field theories as well as theories of quantum space-time.The class of operators may also be useful for the efficient realization of new quantum algorithms.

DOI: 10.22323/1.020.0260

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arXiv:hep-lat/0510043 · Proceedings of XXIIIrd International Symposium on Lattice Field Theory — PoS(LAT2005)

Unitary Evolution on a Discrete Phase Space

Stam Nicolis, Emmanuel G. Floratos

Published Dec 14, 2005• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.22323/1.020.0260

Related papers

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May 3, 20180 citations

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We develop a systematic analysis of the metaplectic semigroup

\mathrm{Mp}_+(d,\mathbb{C})

associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by Hörmander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group

\mathrm{Mp}(d,\mathbb{R})

to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.

L^2(\mathbb R^n)

corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.